Opening it up to competition from other providers isn't the same as privatising the NHS. Privatising means actually selling off NHS assets (e.g. hospitals) to private companies who operate on a for-profit basis. There's basically no way in hell that they'd ever be able to do this - it would be electoral suicide to privatise the NHS in the UK.
Lansley's proposals have two broad goals. The first is to separate the people who commission healtchare (i.e. GPs [family doctors], who refer cases to hospitals) from the providers of the more expensive treatments (i.e. hospitals, to whom patients are referred by GPs). The idea is that hospitals and GPs would no longer be in each others' pockets; GPs would have control over their own budgets, meaning that they could now refer their patients to whichever hospital offers the best quality services for the lowest price. The downside is that GPs are already kind of busy; adding more admin duties to their schedule would probably mean they'd have to hire help from a 3rd party, which might end up costing more than the previous system, where a big trust decided all this stuff.
The second plank is to allow more private providers to offer care, too, but there would be something put in place to prevent private providers from cherry-picking the easiest and most profitable cases. The increase in competition would help drive up standards and lower costs, in the same way that when two supermarkets are built next to each other, they drive down costs and increase choice for the consumer. We'd still have standards and targets that need to be maintained, meaning that quality wouldn't suffer as a consequence of lower price. I honestly can't see a downside to this, though there are always risks and unintended consequences. I think the opposition to this is overblown and irrational. It's a HUGE difference between allowing private providers to provide care, and actually selling off the NHS to private providers. The former is additive: it adds to the total stock of hospital beds, medical procedures, doctors, etc that are in this country. The latter is not additive, it's subsitituive: you swap NHS hospital beds for Private hospital beds. Additive is good!
To me, the plan doesn't seem like a terrible idea. The problem, for me, is with its timing; it would cost a crapload of money initially and risks causing major disruptions to services at a time when everyone's budgets are already being squeezed. The proposals were rushed and the government was criticised for it; they have recently watered the changes down, and it looks increasingly likely that the most radical changes will be scrapped altogether. The worst that would happen with these proposals is that it becomes an expensive waste of time. It's nothing like an actual privatisation of the NHS (which would be a bloody disaster, and something I'd oppose absolutely).
Labour is against the proposals. The Lib Dems are mostly against them, too, but being part of government means they can't freely voice their annoyance as loudly as they would have done in opposition. But really, the reforms proposed weren't significantly different from what Labour was proposing back in 1997; Labour ran into similarly loud and vociferous opposition from doctors and nurses, and similarly had to water them down.
I would be VERY wary about trusting doctors when it comes to NHS reform. They generally act as though they're acting in the best interests of medical care, but actually they have a strong tendency to act as self-servingly as any of the professional classes, like lawyers and bankers. At the end of the day, turkeys don't vote for Christmas; doctors will always fight against changes to the NHS, even when those changes are in their patients' best interest. Notice how they fought tooth and nail for massive pay rises a few years ago, even when other public servants were getting squeezed. They're protecting a closed shop; you shouldn't trust them to voice their opinions on NHS changes any more than you should trust a banker's or lawyer's opinions on changes to the rules surround their professions.
(Full Disclosure: My dad's a doctor and my mum also worked in the NHS all her life.)